The primary reason why people bang up their limbs (arms, legs, elbows, knees, and head) is to be able to lessen the pain of the impact. Secondarily, by lessening the pain, the body is able to make mechanical adaptations to improve the power of the strike.
First with regards to the pain lessening...
While I do believe this does deaden the nervous system's response, I do not think it's generally permanent. It's short-term, probably on the order of a day before the effect starts to wear off. And it lasts maybe a week or two before it goes away completely.
Also, I don't think it's actually damaging the nerves themselves. Probably it's simply causing the brain to reduce its perception of the pain, basically dialing down the "volume" of the pain that the brain hears.
When a nerve in your limbs feels pain, it sends a pain signal to the brain. In the brain, neurotransmitters are responsible for communicating the pain signal to the brain's neurons. You only "experience" the pain at this point.
What happens next is the brain responds by secreting neurotransmitter blockers that prevent the pain chemicals from causing further pain. (The same thing happens when you take an aspirin, by the way.) Repeated doses of pain, will raise the base level of these neurotransmitter blockers so that it takes stronger and stronger amounts of pain for it to be felt by the brain.
And so what you're actually doing when you hit your limbs hard against stuff is increasing the base level of pain blocking chemicals in your brain. That lets you experience less pain. And it wears off over time, if you stop doing the exercises that cause pain.
Unlike taking an aspirin and having it lessen all of the pain you feel in your entire body, this effect is localized to the neurons that are associated with the particular limbs that were hit. And there might be some pain blocking going on right at the source (the limb's nerves) instead of just inside of the brain itself... But at this point, I'm in over my head, since I'm not a neuroscientist. Hehe.
Now, when nerves are seriously damaged (for real), they won't respond to sensation at all. Not just pain, but all sensation including touch and temperature. Your arm, for example, will be numb for weeks after that, unable to even feel normal touches. Or you might feel something, but it won't feel normal. Usually the body can heal minor nerve damage and even reroute signals to the brain to bypass damaged or dead nerves. In worse cases, this damage is more or less permanent, though there might be some sensation on occasion. The body is basically unable to repair it. In the worst cases, the nerve is completely severed, and you feel absolutely nothing ever again.
The moment you damage your nerve, it will sometimes feel like a sharp, shooting electric pain going up the limb, followed by pins and needles and then numbness. Also, when this happens, it's not just pain that will be numbed. It will be all sensation, including touch and temperature.
This is generally NOT what people feel when they perform the "iron body" stuff that martial arts practice. They are still able to feel touch and temperature just fine. It's just that the pain they feel will be lessened. So this is why I say it's probably mostly happening in the brain by increasing levels of neurotransmitter blockers, rather than actually damaging the nerves themselves.
The second part of what's going on is mechanical adaptation. That's where you learn to adjust your body's mechanics to allow you to use the limb to hit harder. If you anticipate that it will be too painful to hit in such a way, you will not really "lean into" your technique. You'll be too timid. And so your mechanics will not be as strong. But when you are sure it's not going to hurt, you start to make those mechanical adaptations to increase the effect and power of your strike.
Now, what I left out was bone strengthening. The theory goes like this: Micro-fractures in bone cause rapid mineralization of the fracture site, filling it up with calcium to reform the bone. So by repeatedly causing these micro-fractures, over time it can lead to a much stronger bone.
But it turns out that might not actually be true. According to some medical doctors, the fractures do increase strength temporarily, but eventually the bone becomes as weak as any other part of the bone, meaning that it's just as likely to break there as any other part of the same bone over the long term: http://www.nytimes.com/2010/10/19/health/19really.html
Of course, that article refers to medical bone breaks and fractures. But what about martial arts micro-fracturing? All bets are off. I don't think it has been sufficiently studied to say either way.
My feeling is that there probably will be some increase in bone strength, but it will only last until you stop the iron body training. At which point, it will probably return to normal strength. Most likely, then, what people perceive as their bones getting stronger is actually just a lessening of pain and an increase in power due to mechanical adaptations as a result (mentioned above).
One of the things you can definitely do to increase bone density and bone volume is to start doing weight lifting exercises. Increasing the load (weight) your bones have to endure will cause your bones to grow. This is described in Wolfe's Law. But once again, this is temporary. Your bone's strength will decrease again after you stop loading it.
Anyway, how important is this "iron body" practice in martial arts? Well, that depends on the style. Styles that do a lot of iron body training probably need it more than styles that practice it less or not at all. Without it, hitting your bones against something will cause a lot of pain, and that will cause you to be timid about using your bones in the way in which your style requires. Some say that's probably a good thing, some obviously would disagree.
Muay Thai, for example, uses the shin in its version of the round kick. They practice by hitting stiff punching bags at first. And then later on they'll work on hitting their shins against harder objects. The object is to lessen pain and make the shin bone stronger. This in turn gives them confidence enough to swing their legs really powerfully without worrying about breaking their shin bones. And in general, it works in their favor. But that's because they typically land their round kicks on their opponent's softer areas, like the thigh muscles and the abdomen. But they often do break their shins when they try the same kind of kick on an opponent who turns his knee into the shin. That happens because the lower part of the tibia (shin bone) is inherently weaker than the top part of it (just below the knee). Probably no amount of toughening exercises can make up for this difference.
So in the end, maybe it's just the pain lessening and mechanical adaptation that's important. The bone strengthening is probably a red herring. Or at the very least, it's overemphasized. In my opinion.
Final word about safety: Obviously, iron body training can cause permanent nerve damage. So work up to it gradually. If you feel shooting pain, pins and needles, or numbness, stop immediately and don't return to it until you know you can handle it and have corrected whatever you were doing wrong. Because, like I said, normally this shouldn't happen. Start slowly and gradually increase force.